2000/10/25 by S. Encinas, A. Nobile, O. Villamayor
Mathematics · #math.AG #msc:14E15 #msc:14D99
published as Proc. London Math. Soc. (3) 86 (2003) 607-648 · Plain TeX, 37 pages
arxiv created 2000/10/25 · arxiv updated 2009/11/30
Given an algorithm of resolution of singularities satisfying certain conditions (``good algorithms''), natural notions of simultaneous algorithmic resolution, or equiresolution, for families of embedded schemes (parametrized by a reduced scheme T) are proposed. It is proved that these conditions are equivalent. Something similar is done for families of sheaves of ideals, here the goal is algorithmic simultaneous principalization. A consequence is that given a family of embedded schemes over a reduced T, this parameter scheme can be naturally expressed as a disjoint union of locally closed sets Tj, such that the induced family on each part Tj is equisolvable. In particular, this can be applied to the Hilbert scheme of a smooth projective variety; in fact, our result shows that, in characteristic zero, the underlying topological space of any Hilbert scheme parametrizing embedded schemes can be naturally stratified in equiresolvable families.